Invertible completions of $2\times 2$ upper triangular operator matrices

In this note we prove that if

is a $2\times 2$ upper triangular operator matrix acting on the Banach space $X\oplus Y$, then $M_{C}$ is invertible for some $C\in \mathcal{L}(Y,X)$ if and only if $A\in \mathcal{L}(X)$ and $B\in \mathcal{L}(Y)$ satisfy the following conditions:

(i)

$A$ is left invertible;

(ii)

$B$ is right invertible;

(iii)

$X/A(X)\cong B^{-1}(0)$.

Furthermore we show that $\sigma (A)\cup \sigma (B)=\sigma (M_{C})\cup W$, where $W$ is the union of certain of the holes in $\sigma (M_{C})$ which happen to be subsets of $\sigma (A)\cap \sigma (B)$.